Appendix B: Simulations to check the kinematic analysis

In Paper I we did numerical simulations in order to evaluate the biases in the kinematic model parameters (in that case, the Oort constants and the solar motion components) induced by our observational constraints and errors. In the present work, we have also generated simulated samples in the same way, though the significant correlations detected between some parameters make it advisable, in this case, to carry out a more detailed study.

In this section we present the process used to generate the simulated samples (the same as in Paper I, except for the change in the systematic contributions considered), the results we obtained and, finally, the quantification of the biases present in our real samples.

B.1 Process used to generate the simulated samples

To take into account the irregular spatial distribution of our stars and their observational errors, parameters describing the position of each simulated pseudo-star were generated as follow:

• From each real star we generated a pseudo-star that has the same nominal position (R₀,l,b) - not affected by errors - as the real one;

• We assumed that the angular coordinates (l,b) have negligible observational errors;

• The distance error of the pseudo-star has a distribution law:
  

  $$\varepsilon(R) = {\rm e}^{ - \frac {1}{2} \left( \frac {R - R_0}{\sigma_R} \right)^2}$$ (B.1)

  
  
  where $\sigma_R$ is the individual error in the photometric distance of the real star (R₀).

To generate the kinematic parameters we randomly assigned to each pseudo-star a velocity (U,V,W) by assuming a cosmic dispersion $(\sigma_U, \sigma_V, \sigma_W)$ and a Schwarzschild distribution:

$$\varphi'_{\rm v}(U, V, W) = {\rm e}^{ - \frac {1}{2} \left( ... ...^2 - \frac {1}{2} \left( \frac {W - W_0}{\sigma_W} \right)^2}$$ (B.2)

where (U₀,V₀,W₀) are the reflex of solar motion. These components were transformed into radial velocities and proper motions in galactic coordinates using the nominal position of the pseudo-star (R₀,l,b). The systematic motion due to galactic rotation and spiral arm kinematics was added following Eqs. (A.4) and (A.19), obtaining the components $(v_{{\rm r_0}}, \mu_{{\rm l_0}}, \mu_{{\rm b_0}})$ for each pseudo-star. Finally, individual observational errors were introduced by using the error function:

$$\varepsilon(v_{{\rm r}}, \mu_{{\rm l}}, \mu_{{\rm b}}) = {\rm... ...rm b}} - \mu_{{\rm b_0}}} {\sigma_{\mu_{{\rm b}}}} \right)^2}$$ (B.3)

where $\sigma_{v_{{\rm r}}}$, $\sigma_{\mu_{{\rm l}}}$ and $\sigma_{\mu_{{\rm b}}}$ are the observational errors of the real star.

At the end of this process we had the following data for each pseudo-star: galactic coordinates (R,l,b), velocity parameters ( $v_{{\rm r}}, \mu_{{\rm l}}, \mu_{{\rm b}}$), errors in the velocity parameters ( $\sigma_{v_{{\rm r}}}, \sigma_{\mu_{{\rm l}}}, \sigma_{\mu_{{\rm b}}}$) and error in the photometric distance ($\sigma_R$). The simulated radial component of those pseudo-stars generated from a real star without radial velocity was not used, thus we imposed on the simulated sample the same deficiency in radial velocity data that is present in our real sample (see Sect. 2.2 and Appendix B in Paper I for more details).

Following this scheme, several sets of 50 simulated samples for both O and B stars and Cepheids were built. A classical solar motion of (U,V,W) = (9,12,7) km s^-1 was considered, taking the dispersion velocity components $(\sigma _U,\sigma _V,\sigma _W) = (8,8,5)$ km s^-1 for O and B stars (see Paper I) and $(\sigma _U,\sigma _V,\sigma _W) = (13,13,6)$ km s^-1 for Cepheids (Luri 2000). For the galactic rotation parameters, we chose the values $a_{{\rm r}} = -2.1$ km s^-1 kpc^-1 and $b_{{\rm r}} = 0.0$ km s^-1 kpc^-2, which correspond to a linear rotation curve with an A Oort constant of 14.0 km s^-1 kpc^-1. On the other hand, for the spiral structure parameters several sets of values were used for $\psi _\odot$ (from $\psi_\odot = 0\hbox{$^\circ$ }$ to $\psi_\odot = 315\hbox{$^\circ$ }$, in steps of $45\hbox{$^\circ$ }$) and $\Omega _{{\rm p}}$ (from $\Omega_{{\rm p}} = 10$ km s^-1 kpc^-1 to $\Omega_{{\rm p}} = 40$ km s^-1 kpc^-1, in steps of 5 km s^-1 kpc^-1), whereas a fixed value of $f_{{\rm r}} = 0.05$ was considered (Yuan 1969). From $(\sigma_U, \sigma_V, \sigma_W)$, $a_{\rm r}$, $\Omega _{{\rm p}}$and $f_{{\rm r}}$, the values of $\Pi _{\rm b}$, $\Theta _{\rm b}$ and $f_\odot$ were inferred for each set of samples.

56 sets of 50 samples for both O and B stars and Cepheids were generated. Concerning the free parameters of our model, in a first stage we adopted classical values (m = 2, $i = -6\hbox{$^\circ$ }$, Lin et al. 1969; $\varpi_\odot = 8.5$ kpc, $\Theta(\varpi_\odot) =220$ km s^-1, Kerr & Lyndell-Bell 1986), though we also tested cases with m = 4, $i = -12\hbox{$^\circ$ }$ (Amaral & Lépine 1997) and $\varpi_\odot = 7.1$ kpc, $\Theta(\varpi_\odot) = 184$ km s^-1 (Olling & Merrifield 1998). In Table B.1 we summarize all the adopted kinematic parameters.

 

 

Table B.1: Parameters of the simulated samples.

$U_\odot$	9 km s-1
$V_\odot$	12 km s-1
$W_\odot$	7 km s-1
$(\sigma_U, \sigma_V, \sigma_W)$	(8,8,5) km s-1 (O and B stars)
	(13,13,6) km s-1 (Cepheid stars)
$a_{\rm r}$	-2.1 km s-1 kpc-1
$b_{\rm r}$	0.0 km s-1 kpc-2
$f_{{\rm r}}$	0.05
$\psi _\odot$	from 0 $\hbox{$^\circ$ }$ to 360 $\hbox{$^\circ$ }$,
	in steps of 45 $\hbox{$^\circ$ }$
$\Omega _{{\rm p}}$	from 10 km s-1 kpc-1 to 40 km s-1 kpc-1,
	in steps of 5 km s-1 kpc-1
Case A	m = 2, $i = -6\hbox{$^\circ$ }$,
	$\varpi_\odot = 8.5$ kpc, $\Theta(\varpi_\odot) =220$ km s-1
Case B	m = 2, $i = -6\hbox{$^\circ$ }$,
	$\varpi_\odot = 7.1$ kpc, $\Theta(\varpi_\odot) = 184$ km s-1
Case C	m = 4, $i = -12\hbox{$^\circ$ }$,
	$\varpi_\odot = 8.5$ kpc, $\Theta(\varpi_\odot) =220$ km s-1
Case D	m = 4, $i = -12\hbox{$^\circ$ }$,
	$\varpi_\odot = 7.1$ kpc, $\Theta(\varpi_\odot) = 184$ km s-1

B.2 Results and discussion

B.2.1 Results for a 2-armed model of the Galaxy

A complete solution simultaneously taking into account radial velocity and proper motion data was computed. Our test showed that the number of Cepheids within 2 kpc from the Sun prevents the obtainment of reliable results. In Figs. B.1 and B.2 we show the results obtained for the simulated samples of O and B stars (0.6 < R <2 kpc) and Cepheids (0.6 < R < 4 kpc) in case A (see Table B.1).

Figure B.1: Bias (obtained value-simulated value; left) and standard deviation (right) for each set of 50 samples in the solar motion (top), galactic rotation (middle) and spiral arm kinematic (bottom) parameters for O and B pseudo-stars (Case A). Values of $\Omega _{{\rm p}}$: 10 km s-1 kpc-1 (black double solid line), 15 (black solid line), 20 (grey solid line), 25 (dotted line), 30 (dashed line), 35 (long dashed line) and 40 (dot-dashed line).

Figure B.2: Bias (obtained value-simulated value; left) and standard deviation (right) for each set of 50 samples in the solar motion (top), galactic rotation (middle) and spiral arm kinematic (bottom) parameters for Cepheid pseudo-stars (Case A). Values of $\Omega _{{\rm p}}$: 10 km s-1 kpc-1 (black double solid line), 15 (black solid line), 20 (grey solid line), 25 (dotted line), 30 (dashed line), 35 (long dashed line) and 40 (dot-dashed line).

As a first conclusion, and confirming our suspicions, a systematic trend with $\psi _\odot$ and/or $\Omega _{{\rm p}}$ is observed in most cases. This behaviour is produced by the correlations between some terms in the least squares fit, which depends on the spatial distribution of each sample.

For solar motion a bias between -1.5 and 1.5 km s^-1 (depending on $\psi _\odot$ and $\Omega _{{\rm p}}$) was found for $U_\odot$ and $V_\odot$, and of only -0.3 km s^-1 for $W_\odot$. For both O and B stars and Cepheids we found the bias on $V_\odot$ and $W_\odot$ to be independent of $\Omega _{{\rm p}}$, with a slight dependence on $\psi _\odot$. For $\psi_\odot = 270\hbox{$^\circ$ }$ and $\Omega_{{\rm p}} = 10$ km s^-1 kpc^-1 we found a large negative bias on $V_\odot$, but with a great standard deviation. This occurred in several samples (inside this set) with serious convergence problems in the iteration procedure we use to solve the least squares fit. Similar problems in other cases with $\psi_\odot = 270\hbox{$^\circ$ }$ will be found later. For O and B stars, the standard deviations in the solar motion components are $\approx$0.6 km s^-1 for $U_\odot$ and $V_\odot$ (except for $\psi_\odot = 270\hbox{$^\circ$ }$), and $\approx$0.3 km s^-1 for $W_\odot$. On the other hand, in the case of Cepheids these values increased to $\approx$1.4-2.0 km s^-1and $\approx$0.8 km s^-1, respectively.

The biases found in the first- and second-order terms of the galactic rotation curve are negligible for Cepheids, with a level fluctuation of $\pm 0.3$ km s^-1 kpc^-1 (or km s^-1 kpc^-2). In this case there is a standard deviation of 1.3 for $a_{\rm r}$ and 0.5 for $b_{\rm r}$. For O and B stars the biases clearly depend on $\psi _\odot$, varying from -0.7 to -0.5 km s^-1 kpc^-1 for $a_{\rm r}$, and from -1.0 to 0.1 km s^-1 kpc^-2 for $b_{\rm r}$. The standard deviations are 0.8 km s^-1 kpc^-1and 1.2 km s^-1 kpc^-2, respectively.

Let us study the biases that have an effect on the determination of the spiral structure parameters. As a general conclusion, Figs. B.1 and B.2 show that our sample of O and B stars supplies better results than the Cepheid sample.

In the case of O and B stars, we found a clear dependence with $\psi _\odot$ and $\Omega _{{\rm p}}$ in $\psi _\odot$, $\Pi _{\rm b}$ and $\Theta _{\rm b}$ determinations, whereas $f_\odot$ and $\Omega _{{\rm p}}$ only show peculiar behaviour around $\psi_\odot = 270\hbox{$^\circ$ }$. Concerning $\psi _\odot$, the bias oscillates from $-20\hbox{$^\circ$ }$ to $30\hbox{$^\circ$ }$. The standard deviation of the mean for the 50 samples of each set is about 10- $20\hbox{$^\circ$ }$. On the other hand, for the amplitudes $\Pi _{\rm b}$ and $\Theta _{\rm b}$ the biases are of $\pm 2$ km s^-1, with a standard deviation of about 1 km s^-1. Neither $f_\odot$ nor $\Omega _{{\rm p}}$ have a considerable bias, except for $\psi_\odot = 270\hbox{$^\circ$ }$, where both biases and standard deviations go up.

For Cepheid stars similar results were obtained, but with larger standard deviations in all cases. The bias in $\psi _\odot$ changes from $-20\hbox{$^\circ$ }$to $10\hbox{$^\circ$ }$, with standard deviations of about 30- $60\hbox{$^\circ$ }$. In the case of $\Pi _{\rm b}$, we found a clear dependence on both $\psi _\odot$ and $\Omega _{{\rm p}}$, with a bias of $\pm1.5$ km s^-1 and a standard deviation of about 2 km s^-1. On the other hand, for $\Theta _{\rm b}$ the bias is smaller, from 0 to 0.8 km s^-1, and the standard deviation is 1-1.5 km s^-1. As for O and B stars, for $f_\odot$ and $\Omega _{{\rm p}}$ small biases are found, though the standard deviations are larger in this case.

B.2.2 Results considering possible errors in the choice of the free parameters

An interesting point to analyse is the study of the biases produced by a bad choice of the free parameters in our model (m, i, $\varpi_\odot$, $\Theta (\varpi _\odot )$). In the same way as in the previous section, we simulated 50 samples for each one of the cases considered in the real resolution, i.e. cases A, B, C and D (see Table B.2). The simulated parameters were the same as in Table B.1 for solar motion and galactic rotation. For spiral arm kinematics, we considered $\psi_\odot = 315\hbox{$^\circ$ }$ and $\Omega_{{\rm p}} = 30$ km s^-1kpc^-1 (similar values to those obtained from real samples; see Sect. 6).

 

 

Table B.2: Bias and standard deviation in $\psi _\odot$ and $\Omega _{{\rm p}}$ for crossed solutions for the simulated samples of O and B stars and Cepheids. Units: $\psi _\odot$ in degrees; $\Omega _{{\rm p}}$ in km s^-1 kpc^-1.

	O and B stars with 0.6 < R < 2 kpc				Cepheid stars with 0.6 < R < 4 kpc
	Case A	Case B	Case C	Case D	Case A	Case B	Case C	Case D
	Case A simulated
$\Delta \psi_\odot$	23.	17.	24.	19.	-11.	0.	-2.	3.
$\sigma_{\psi_\odot}$	28.	27.	25.	25.	65.	61.	71.	71.
$\Delta \Omega_{{\rm p}}$	2.2	3.1	-1.1	-0.8	-4.3	-3.5	-2.5	-4.0
$\sigma_{\Omega_{{\rm p}}}$	6.2	7.3	2.9	3.5	18.9	16.6	8.8	7.3
	Case B simulated
$\Delta \psi_\odot$	32.	24.	32.	25.	0.	4.	-11.	1.
$\sigma_{\psi_\odot}$	34.	32.	31.	30.	83.	75.	82.	78.
$\Delta \Omega_{{\rm p}}$	0.7	1.9	-1.6	-1.3	-4.3	-14.3	-4.3	-4.1
$\sigma_{\Omega_{{\rm p}}}$	7.1	8.0	3.8	3.9	16.2	52.8	7.0	8.1
	Case C simulated
$\Delta \psi_\odot$	18.	14.	17.	14.	4.	2.	-2.	10.
$\sigma_{\psi_\odot}$	24.	23.	21.	21.	62.	58.	60.	63.
$\Delta \Omega_{{\rm p}}$	6.7	8.2	1.2	2.0	-2.4	-1.1	-1.0	-3.0
$\sigma_{\Omega_{{\rm p}}}$	6.5	7.3	2.9	3.3	25.1	21.5	10.9	9.2
	Case D simulated
$\Delta \psi_\odot$	29.	23.	25.	21.	9.	10.	-9.	1.
$\sigma_{\psi_\odot}$	31.	29.	27.	26.	84.	78.	81.	73.
$\Delta \Omega_{{\rm p}}$	4.3	6.3	0.2	1.1	-3.9	-4.9	-3.2	-2.8
$\sigma_{\Omega_{{\rm p}}}$	7.1	8.2	3.7	3.8	17.1	19.5	7.8	8.6

In Table B.2 we show the biases and standard deviations when solving the model equations in crossed solutions (e.g. we generated 50 simulated samples considering the free parameters in case A, and then we solved equations using the free parameters adopted for cases A, B, C and D, and so on for the other cases). As a first conclusion, we can observe that a bad choice in the free parameters does not substantially alter the derived kinematic parameters, particularly $\psi _\odot$. In other words, for each set of simulated samples we obtained nearly the same values for the parameters whether we solved the Eqs. (8) with the correct set of free parameters or with a wrong combination of them. Differences in $\psi _\odot$ do not exceed 10 $\hbox{$^\circ$ }$ for O and B stars and 20 $\hbox{$^\circ$ }$ for Cepheids. In the case of $\Omega _{{\rm p}}$ we found large differences in some cases, but always when the standard deviation was also large. This is especially true for Cepheids. A remarkable point is that the minimum bias was not always produced when we properly chose the free parameters.

B.2.3 Conclusions

In the light of these results, we conclude that we are able to determine the kinematic parameters of the proposed model of the Galaxy from the real star samples described in Sect. 2, supposing that the velocity field of the stars is well described by this model. We studied case A (m = 2, $i = -6\hbox{$^\circ$ }$, $\varpi_\odot = 8.5$ kpc, $\Theta(\varpi_\odot) =220$ km s^-1 in detail in these simulations, but we also looked at the other combinations of the free parameters (cases B, C and D), with similar conclusions. Nevertheless, the study of crossed solutions has shown that it will be very difficult to decide between the several set of free parameters discussed in Sect. 6 (see also Table B.2), owing to the small differences obtained when changing the free parameters in the condition equations.